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On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve

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Abstract

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form \(y^q+y=x^m,\,q\) being a prime power and \(m\) a positive integer which divides \(q+1\). The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.

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Notes

  1. Here the subscript red denotes the reduction of a zero-dimensional scheme.

  2. We recall that \(\text{ Res }_L (A \cup E) = \text{ Res }_L (E)\), because \(A\subseteq L\), and \(\deg (\text{ Res }_L (E)) = \deg (E) -\deg (L \cap E)\).

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Acknowledgments

The authors would like to thank the Referee for useful suggestions and comments that improved the presentation of this work.

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Authors and Affiliations

  1. Department of Mathematics, University of Trento, Via Sommarive 14, 38123 , Povo, Trento, Italy

    Edoardo Ballico

  2. Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000 , Neuchâtel, Switzerland

    Alberto Ravagnani

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  1. Edoardo Ballico
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  2. Alberto Ravagnani
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Correspondence to Edoardo Ballico.

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Ballico, E., Ravagnani, A. On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve. AAECC 24, 343–354 (2013). https://doi.org/10.1007/s00200-013-0206-z

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  • DOI: https://doi.org/10.1007/s00200-013-0206-z

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